Optimal Power Flow with Stochastic Solar Power Using Clustering-Based Multi-Objective Differential Evolution

Abstract

Optimal power flow is one of the fundamental optimal operation problems for power systems. With the increasing scale of solar energy integrated into power systems, the uncertainty of solar power brings intractable challenges to the power system operation. The multi-objective optimal power flow (MOOPF) considering the solar energy becomes a hotspot issue. In this study, a MOOPF model considering the uncertainty of solar power is proposed. Both scenarios of overestimation and underestimation of solar power are modeled and penalized in the form of operating cost. In order to solve this multi-objective optimization model effectively, this study proposes a clustering-based multi-objective differential evolution (CMODE) which is based on the main features: (1) extending DE into multi-objective algorithm, (2) introducing the feasible solution priority technique to deal with different constraints, and (3) combining the feasible solution priority technique and the merged hierarchical clustering method to determine the optimal Pareto frontier. The simulation outcomes on two cases based on the IEEE 57-bus system verify the reliability and superiority of CMODE over other peer methods in addressing the MOOPF.

1. Introduction

Optimal power flow (OPF) is a typical operation problem of power systems. The traditional OPF usually focus on minimizing the overall operating cost of thermal power generators. However, thermal power generators release massive pollutants during operation. Therefore, emission reduction has become one of the most notable hotspot issues. Under this background, power systems need to minimize the emission of pollutants while operating economically, which is also the purpose of the studied multi-objective optimal power flow (MOOPF).

MOOPF is a highly nonconvex and nonlinear multi-objective problem, especially with the valve-point effects of thermal generators. Linear programming [1] and dynamic programming [2] are representative classical methods for addressing the problem, but they are vulnerable to the issue of stagnation when applied to complicated problems [3]. Furthermore, they generally need to transform multiple objectives into one, so they cannot handle all the objectives at the same time and provide a group of compromise solutions when solving multi-objective problems.

Intelligent methods, as the most favorite approach in recent years, can optimize multiple objectives simultaneously without transforming these objectives. Moreover, they do not impose any form of restriction on the problem. In order to solve the MOOPF, Pulluri et al. [4] raised the ability of different evolution through a combination of eigenvector and binomial crossovers and self-adaptive parameter strategy. El-Sattar et al. [5] proposed a multi-objective Jaya algorithm to deal with this issue. Chen et al. [6] introduced nonlinearly-adjusted coefficient, adaptive-adjusted map factor and penalty function into pigeon-inspired algorithm and combined a sorting rule to solve the problem. Naderi et al. [7] combined fuzzy strategy, differential evolution and self-adaptive particle swarm optimization to address the problem. Reddy [8] employed a multi-objective differential evolution to enhance the solving reliability and efficiency for MOOPF. Abbasi et al. [9] integrated an effective initialization method and differential evolution into harmony search. Biswas et al. [10] combined the feasible solution technology with classical decomposition based multi-objective evolutionary algorithm to address this complex problem.

In solving practical complex multi-objective engineering problems such as the MOOPF, achieving the optimal Pareto frontier (PF) is one of the most difficult issues. In order to solve this issue, Xiong et al. [11] proposed an improved correction strategy by combining the slope and the crowding distance to obtain a better PF. Warid et al. [12] presented a quasi-oppositional Jaya and used an external elitist repository to produce a well-distributed PF. Shi et al. [13] combined some diversity maintaining strategies to obtain a smooth PF. Hua et al. [14] proposed a clustering-based environment selection to address the issue. Li et al. [15] adopted the reinforcement learning to guide the agents toward a better PF. Zhang et al. [16] raised a non-dominated improved multi-objective teaching-learning-based optimization by introducing an adaptive teaching factor and a non-dominated ranking method based on crowding distance to find the best PF.

Solar photovoltaic (PV) power can help power systems achieve the purpose of energy saving and emission reduction significantly [17,18,19]. Nevertheless, with the increasing scale of PV power, its stochastic output [20,21] brings many challenges to the operation of power systems. Moreover, this strong uncertainty also makes the MOOPF more complex and the optimal PF is harder to be searched. In order to achieve a good PF for a complex multi-objective problem, the clustering-based technique is a workable way. The clustering-based technique, such as partitioning-based clustering, grid-based clustering, model-based clustering and density-based clustering, can scan solutions and help them approach a better PF. However, these clustering techniques also have some weaknesses [14]. For instance, the model-based clustering is complex and takes a long time. The grid-based clustering cannot be employed independently. The partitioning-based clustering needs to set up the cluster centers in advance. The density-based clustering is more likely to choose the individuals in the crowded regions, but some individuals in the sparse regions may be better.

In contrast with the above clustering-based techniques, the merging hierarchical clustering-based technique [22] does not require assumptions about the distribution of individuals. It also does not require the number of clusters to be set in advance. Inspired by the potential of the merging hierarchical clustering-based technique, we propose a multi-objective optimization algorithm named CMODE by hybridizing it with differential evolution (DE) and apply the resultant CMODE to the MOOPF considering the uncertainty of PV power in this work. Firstly, we extend DE to adapt to the multi-objective problem. Secondly, to address the MOEED reliably, we adopt the feasible solution priority technique to process different constraints. Thirdly, we incorporate the merging hierarchical clustering-based technique and the feasible solution priority technique to select compromise solutions to acquire the optimal PF.

The main contributions of this study are concluded below:

(i) A MOOPF model with the PV power uncertainty is presented. A method based on overestimating and underestimating the PV power output is introduced to processing the uncertainty.

(ii) A multi-objective method named CMODE is developed to solve the MOOPF model. Some approaches including fuzzy decision technique, feasible solution priority technique and merging hierarchical clustering are integrated into CMODE to raise its solving performance.

(iii) In order to test the usefulness of CMODE, two cases based on the IEEE 57-bus system are conducted. Furthermore, three advanced methods are selected to compare with CMODE.

The rest structure can be summarized as below. Section 2 introduces the studied MOOPF and the calculation of stochastic PV power. Section 3 briefly reviews the DE and the merged hierarchical clustering. Section 4 provides the proposed CMODE. Section 5 gives the testing results and Section 6 summarizes the conclusions and future work.

2. Mathematical Model of the Studied MOOPF

MOOPF is a complex optimization problem, mainly by adjusting some control variables to minimize the optimization objectives of power system while satisfying the operating constraints [23]. The optimization objectives of the studied MOOPF can be described as below:

$$Minimize:Y(p,o)=[E(p,o),C(p,o)]$$

(1)

$$subject to\{\begin{array}{c}g(p,o)\le 0\\ h(p,o)=0\end{array}$$

(2)

where $p$ and $o$ are the vectors of control and state variables, respectively; $Y(p,o)$ means the objective functions; $E(p,o)$ means the total emission; $C(p,o)$ means the total operating cost; $g(p,o)$ means the set of inequality constraints; $h(p,o)$ means the set of equality constraints.

The group of control variables is used to control the system power flow and it can be described as below:

$$p=\left\{P_{G2},P_{G3},\dots ,P_{G{N}_G},V_{G1},V_{G2},\dots ,V_{G{N}_G},T_1,T_2,\dots ,T_{N_{tr}}\right\}$$

(3)

where $P_{Gm}$ and $V_{Gm}$ are the active power and voltage of generator on bus $m$, respectively; $T_m$ is the tap of $m$th transformer; $N_G$ and $N_{tr}$ are the number of generators and tap-regulated transformers, respectively.

The group of state variables is used to define the system state and it can be described as below:

$$o=\left\{P_{G1},V_{LB1},V_{LB2},\dots ,V_{LB{N}_{LB}},Q_{G1},Q_{G2},\dots ,Q_{G{N}_G},S_{L1},S_{L2},\dots ,S_{L{N}_L}\right\}$$

(4)

where $P_{G1}$ is the active power of generator at swing bus; $V_{LBm}$ is the voltage of $m$th load bus; $Q_{Gm}$ is the reactive power of generator on bus $m$; $S_{Lm}$ is the line loading of $m$th branch; $N_{LB}$ and $N_L$ are the number of load buses and branches, respectively.

2.1. Inequality and Equality Limits

(1)

Generator limits

$$Q_{Gm}^{\min }\le Q_{Gm}\le Q_{Gm}^{\max }$$

(5)

$$V_{Gm}^{\min }\le V_{Gm}\le V_{Gm}^{\max }$$

(6)

$$P_{Gm}^{\min }\le P_{Gm}\le P_{Gm}^{\max }$$

(7)

where $Q_{Gm}^{\min }$ and $Q_{Gm}^{\max }$ are the upper and lower reactive power limit of the $m$th generator, respectively; $V_{Gm}^{\min }$ and $V_{Gm}^{\max }$ are the upper and lower voltage limit of the $m$th generator, respectively; $P_{Gm}^{\min }$ and $P_{Gm}^{\max }$ are the upper and lower active power limit of the $m$th generator, respectively.

(2)

Transformer limits

$$T_m^{\min }\le T_m\le T_m^{\max }$$

(8)

where $T_m^{\min }$ and $T_m^{\max }$ are the upper and lower limit of the $m$th transformer, respectively.

(3)

Operation limits

$$V_{Bm}^{\min }\le V_{Bm}\le V_{Bm}^{\max }$$

(9)

$$S_{Lm}\le S_{Lm}^{\max }$$

(10)

where $V_{Bm}$ is the voltage on bus $m$; $V_{Bm}^{\min }$ and $V_{Bm}^{\max }$ are the upper and lower voltage limit of the $m$th bus, respectively; $S_{Lm}^{\max }$ is the upper limit of the capacity of branch $m$.

(4)

Power balance limits

$$P_{Gm}=V_{Bm}{\displaystyle \sum _{n=1}^{N_B}{V}_{Bn}\left[B_{mn}\sin \left({\delta }_m-{\delta }_n\right)+G_{mn}\cos \left({\delta }_m-{\delta }_n\right)\right]}+P_{Lm}$$

(11)

$$Q_{Gm}=V_{Bm}{\displaystyle \sum _{n=1}^{N_B}{V}_{Bn}\left[G_{mn}\cos \left({\delta }_m-{\delta }_n\right)-B_{mn}\sin \left({\delta }_m-{\delta }_n\right)\right]}+Q_{Lm}$$

(12)

where ${\delta }_m$ is the voltage angle on bus $m$; $B_{mn}$ and $G_{mn}$ are the susceptance and conductance between buses $m$ and $n$, respectively; $N_B$ is the number of buses; $P_{Lm}$ and $Q_{Lm}$ are the active and reactive load demand on bus $m$, respectively.

2.2. Operating Cost of Stochastic PV Power

2.2.1. Power Model of PV Power Plants

The power output of a PV power plant $P_V$ varies with the solar irradiance $R_I$, and the output power is calculated as below:

$$P_V\left(R_I\right)=P_S\left(R_I/R_{I.std}\right)$$

(13)

where $R_{I.std}$ is the stands for the solar irradiance under the normal condition; $P_S$ is the rated power of PV power plant.

In addition, the solar irradiance generally obeys the Lognormal probability density function $F\left(R_I\right)$, as given below [24]:

$$F\left(R_I\right)=\exp \left[\frac{-{\left(\ln \left(R_I\right)-\phi \right)}^2}{2{\gamma }^2}\right]/R_I\gamma \sqrt{2\pi }$$

(14)

where $\phi$ and $\gamma$ are the mean and standard deviation in Lognormal probability density function, respectively.

The distribution of $S_I$ obtained by running 12,000 Monte Carlo scenes under $\phi =5.2$ and $\gamma =0.6$ is illustrated in Figure 1.

2.2.2. Total Operating Cost of PV Power Plants

The output forecasting of a PV power plant has both possibilities of underestimation and overestimation [16]. Thus, in addition to the direct cost, the total operating cost also contains the cost for underestimation and overestimation scenarios.

The direct cost $C_{Vdm}\left(P_{Vm}\right)$ is given by:

$$C_{Vdm}\left(P_{Vm}\right)={\varpi }_d{P}_{Vm}$$

(15)

where $P_{Vm}$ is the estimated power of $m$th PV power plant; ${\varpi }_d$ is the direct cost factor.

The reserve cost $C_{Vrm}\left(P_{Vm}\right)$ due to the underestimation is given by:

$$C_{Vrm}\left(P_{Vm}\right)={\varpi }_r{\displaystyle \sum _{n=1}^{N_{n-}}\left[\left(P_{Vm}-P_{n-}\right)f_{n-}\right]}$$

(16)

where ${\varpi }_r$ is the reserve cost factor; $P_{n-}$ is the active power lower than $P_{Vm}$; $f_{n-}$ is the relative frequency of $P_{n-}$; $N_{n-}$ is the number of $P_{n-}$.

The penalty cost $C_{Vpm}\left(P_{Vm}\right)$ due to the overestimation is given by:

$$C_{Vpm}\left(P_{Vm}\right)={\varpi }_p{\displaystyle \sum _{n=1}^{N_{n+}}\left[\left(P_{n+}-P_{Vm}\right)f_{n+}\right]}$$

(17)

where ${\varpi }_p$ is the penalty cost factor; $P_{n+}$ is the active power higher than $P_{Vm}$; $f_{n+}$ is the relative frequency of $P_{n+}$; $N_{n+}$ is the number of $P_{n+}$.

Thus, the total cost $C_V$ of $N_V$ PV power plants is given by:

$$C_V={\displaystyle \sum _{m=1}^{N_V}\left(C_{Vdm}\left(P_{Vm}\right)+C_{Vrm}\left(P_{Vm}\right)+C_{Vpm}\left(P_{Vm}\right)\right)}$$

(18)

2.3. Total Operating Cost of Thermal Generators

Thanks to the valve point effect, the operating curve of the thermal generator is highly nonlinear. The total operating cost $C_T$ of $N_T$ thermal generators is as below:

$$C_T={\displaystyle \sum _{m=1}^{N_T}\left\{{\eta }_m{P}_m^2+{\mu }_m{P}_m+{\alpha }_m+\left|l_m\times \sin \left[k_m\times \left(P_m-P_m^{\min }\right)\right]\right|\right\}}$$

(19)

where $P_m$ and $P_m^{\min }$ are the output and min power of $m$th thermal generator, respectively; ${\eta }_m$, ${\mu }_m$ and ${\alpha }_m$ are the operating cost factors of $m$th thermal generator; $l_m$ and $k_m$ are the valve-point effect factors of $m$th thermal generator.

2.4. Total Emission of Thermal Generators

The pollutant gases emitted by the thermal generators vary with their power outputs. The total emission $E_T$ is as below:

$$E_T={\displaystyle \sum _{m=1}^{N_T}\left[e_m{P}_m^2+d_m{P}_m+c_m+u_m\exp \left(g_m{P}_m\right)\right]}$$

(20)

where $e_m$, $d_m$, $c_m$, $u_m$ and $g_m$ are the emission factors of $m$th thermal generator.

2.5. Optimization Objectives of the Studied MOOPF

The purpose of the studied MOOPF considering the PV uncertainty is to minimize both the overall emission and operating cost, which is as below:

$$Minimize:[E_T,C_T+C_V]$$

(21)

2.6. Operation Indicators

Two indicators including the total line power losses $P_{loss}$ and voltage deviations $V_D$ are employed to measure the operation quality.

$$P_{loss}={\displaystyle \sum _{L=1}^{N_L}{G}_{L\left(mn\right)}}\left[V_{Bn}^2-2V_{Bn}{V}_{Bm}\cos \left({\delta }_n-{\delta }_m\right)+V_{Bm}^2\right]$$

(22)

$$V_D={\displaystyle \sum _{m=1}^{N_B}\left|1-V_{Bm}\right|}$$

(23)

where $N_L$ is the number of branches; $G_{L\left(mn\right)}$ is the conductance of branch $L$ connecting buses $m$ and $n$.

3. Differential Evolution and Merged Hierarchical Clustering

3.1. Differential Evolution

DE is an advanced intelligent method that mainly contains four operators [25,26,27,28].

(1) Initialization. The optimization process starts from an initial population $Init\_pop$ with a set of individuals and each one is initialized as below:

$$x_{m,n}^0=x_{m,n}^{\min }+rand(0,1)\times (x_{m,n}^{\max }-x_{m,n}^{\min })$$

(24)

where $m\in \left[1,N\right]$, $N$ means the population size; $n\in \left[1,D\right]$, $D$ means the problem scale; $rand(0,1)$ is a random value between 1 and 0; $x_{m,n}^{\min }$ and $x_{m,n}^{\max }$ mean the lower and upper limits, respectively.

Therefore, the initial population can be expressed as below:

$$Init\_pop=\left\{x_1^G,x_2^G,\dots ,x_N^G\right\}$$

(25)

where $G$ means the iteration counter.

(2) Mutation. It is to generate a mutant vector $v_m^G$ from the target individual $x_m^G$ by:

$$v_m^G=x_{r1}^G+F(x_{r2}^G-x_{r3}^G)$$

(26)

where $F$ is the scaling factor; $x_{r1}^G$, $x_{r2}^G$ and $x_{r3}^G$ are three random distinct individuals.

(3) Crossover. Its intention is to combine $x_m^G$ and $v_m^G$ to produce a new vector $u_m^G$ by:

$$u_{m,n}^G=\{\begin{array}{l}{v}_{m,n}^G, \mathrm{if} n=n_{rand} \mathrm{or} \mathrm{rand}(0,1)\le CR\\ x_{m,n}^G, \mathrm{otherwise}\end{array}$$

(27)

where $n_{rand}$ is a random integer within $\left[1,D\right]$; $CR$ denotes the crossover rate.

(4) Selection. Its purpose is to select those superior individuals for the next iteration. Therefore, the more advantageous individual in $x_m^G$ and $u_m^G$ is retained, as shown below:

$$x_m^{G+1}=\{\begin{array}{l}{u}_m^G, \mathrm{if} Y(u_m^G)\le Y(x_m^G)\hfill \\ x_m^G, \mathrm{otherwise} \hfill \end{array}$$

(28)

where $Y(·)$ means the fitness value.

3.2. Merged Hierarchical Clustering

In the merged hierarchical clustering, all points in a search space are treated as clusters [22]. These clusters are merged consecutively by a specific conjunction standard until they are grouped to one or the number of clusters reaches the predetermined value. The dendrogram of the merged hierarchical clustering is shown in Figure 2. The line between two different points means that they are merged into a new cluster by a special linkage. In order to divide the solutions as evenly as possible into the desired number of groups, we use the Ward’s linkage [29] as the conjunction criterion. The Ward’s linkage maximizes the distance index $\tau \left(m,n\right)$ between different clusters, and minimizes the $\tau \left(m,n\right)$ within the same cluster. The calculation of the $\tau \left(m,n\right)$ is as below:

$$\tau \left(m,n\right)={\Vert {\overline{U}}_m-{\overline{U}}_n\Vert }_2\sqrt{2N_{Um}{N}_{Un}/\left(N_{Um}+N_{Un}\right)}$$

(29)

where ${\Vert ·\Vert }_2$ means Euclidean distance; ${\overline{U}}_m$ and ${\overline{U}}_n$ mean the centers of the cluster $m$ and the cluster $n$, respectively; $N_{Um}$ and $N_{Un}$ mean the total numbers of points in the clusters $m$ and $n$, respectively.

Here, we employ a bi-objective optimization problem to illustrate. Firstly, all points are treated as a single cluster and the $\tau \left(m,n\right)$ between every two points is calculated. Secondly, two points that have the smallest $\tau \left(m,n\right)$ are selected to merge into a new cluster. For instance, suppose $\tau \left(A,B\right)$ is the smallest, then A and B will be assigned into a new cluster G and the new cluster center is as below:

$${\overline{U}}_g=\left(\frac{{\displaystyle {\sum }_m^{N_{Ug}}{Y}_1^{\ast }\left(x_m\right)}}{N_{Ug}},\frac{{\displaystyle {\sum }_m^{N_{Ug}}{Y}_2^{\ast }\left(x_m\right)}}{N_{Ug}}\right)$$

(30)

For robust clustering, all objective values need to be normalized, as given below:

$$Y_j^{\ast }\left(x_m\right)=\left(Y_j\left(x_m\right)-Y_j^{\min }\right)/\left(Y_j^{\max }-Y_j^{\min }\right)$$

(31)

where $Y_j^{\ast }(·)$ means the jth normalized objective value; $Y_j^{\max }$ and $Y_j^{\min }$ are the maximum and minimum values of the jth objective value.

This process will repeat until the terminal condition is met, i.e., the required number of the clusters is satisfied. As shown in Figure 1, different layers contain different clusters. For example, layer 1 contains five clusters, i.e., C, D, E, F and G, while layer 6 just contains one cluster K, so we can obtain the required number of clusters by setting the terminal condition.

4. Clustering-Based Multi-Objective Differential Evolution

4.1. Framework of CMODE

The flowchart of CMODE is shown in Figure 3. Firstly, a random population named Parent is first created and the fitness and the constraint violation values of each member in Parent are evaluated. Secondly, a population named Child is generated by crossing the target individuals and the mutant vectors. Next, the Parent and the Child are combined and the top N individuals are selected to become the new Parent. After that, the merged hierarchical clustering is performed on feasible solutions to achieve the PF. The above steps are executed until the stopping condition is met.

4.2. Feasible Solution Priority Technique

In order to select the solutions that satisfy different constraints, we employ the feasible solution priority technique to address the problem. The technique uses a violation index $\upsilon \left(x\right)$ to judge the feasibility of solutions, as given below [30]:

$$\upsilon \left(x\right)={\displaystyle \sum _{w=1}^{N_W}\frac{I_w\left(x\right)}{I_w^{\max }}}/{\displaystyle \sum _{w=1}^{N_W}\frac{1}{I_w^{\max }}}$$

(32)

where $N_W$ means the number of constraints; $I_w$ means the violation of the wth constraint; $I_w^{\max }$ means the maximum value of $I_w$.

For two different solutions $x_m$ and $x_n$, $x_m$ is preferable to $x_n$ according to the following criteria.

(1) Criterion 1: if $\upsilon \left(x_m\right)=0$ and $\upsilon \left(x_n\right)\ne 0$.

(2) Criterion 2: if $\upsilon \left(x_m\right)=0$, $\upsilon \left(x_n\right)=0$, and $Y(x_m)\le Y(x_n)$.

(3) Criterion 3: if $\upsilon \left(x_m\right)\ne 0$, $\upsilon \left(x_n\right)\ne 0$, and $Y(x_m)\le Y(x_n)$.

4.3. Merged Hierarchical Clustering-Based Determination of PF

Since the number of feasible solutions may exceed the required value N, we propose to application of the merged hierarchical clustering to select superior solutions to form the PF, as shown in Algorithm 1.

We first combine the Parents of two adjacent generations to form a set U, and execute the fast non-dominated sort [31] on U to rank its members into different fronts. If the total amount of members in U is equal to N, then all the members are selected to form the PF. Otherwise, if the total number members in U is higher than N, and the solutions in the first K fronts are less than N while the solutions in the first (K + 1) fronts are higher than N, then we first retain all the solutions in the first K fronts in the PF. For the rest solutions, we make up them from the (K + 1) front by using the merged hierarchical clustering.

In addition, to avoid the situation that some clusters contain no members while some clusters contain four or more members, we use the adjusting strategy in [14] to assign solutions to different clusters. In this way, the members in different clusters are distributed relatively evenly, which make the PF more uniform.

Algorithm 1: Merged hierarchical clustering-based determination of PF

Input: (gen − 1)th Parent and genth Parent

Output: PF

1: Combine the (gen − 1)th Parent and the genth Parent to for a set U.

2: Rank the solutions in U by executing the fast non-dominated sort

% PFK means the solutions in the first K fronts of U

% |PFK| means the capacity of PFK

3: if |PFK| = N then

4: PF = U, break

5: else

6: Select the rest R (|R| = (N − |PFK|)) solutions from the (K + 1) front of U with the merged hierarchical clustering

7: PF = PFK ∪ R

8: end if

4.4. The Determination of the Best Compromise Solution

The fuzzy decision technique is usually employed to extract the best compromise solution based on an index $\zeta \left(x_m\right)$ [25]:

$$\zeta \left(x_m\right)={\displaystyle \sum _{\mathrm{j}=1}^{N_M}{\zeta }_j\left(x_m\right)}/{\displaystyle \sum _{m=1}^N{\displaystyle \sum _{\mathrm{j}=1}^{N_M}{\zeta }_j\left(x_m\right)}}$$

(33)

where $N_M$ means the number of objectives.

The best compromise solution has the highest $\zeta \left(x_m\right)$. In Equation (32), the calculation of ${\zeta }_j\left(x_m\right)$ is as below:

$${\zeta }_j\left(x_m\right)=\{\begin{array}{ll}1& Y_j\left(x_m\right)\le Y_j^{\min }\\ \frac{Y_j^{\max }-Y_j\left(x_m\right)}{Y_j^{\max }-Y_j^{\min }}& Y_j^{\min }\le Y_j\left(x_m\right)\le Y_j^{\max }\\ 0& Y_j\left(x_m\right)\ge Y_j^{\max }\end{array}$$

(34)

5. Simulation Results

5.1. Study Cases

CMODE is implemented to two different IEEE 57-bus systems and the systems’ data are shown in

Table 1. The system information.

Items	Case1 IEEE 57-Bus System		Case2 Modified IEEE 57-Bus System
	Quantity	Details	Quantity	Details
Buses	57	-	57	-
Branches	80	-	80	-
Thermal generators	7	Buses: 1 (Balancing Node), 2, 3, 6, 8, 9 and 12	4	Buses: 1 (Balancing Node), 3, 8 and 12
PV power plant	0	-	3	Buses: 2, 6 and 9
Transformer	17	Branches: 19, 20, 31, 35, 36, 37, 41, 46, 54, 58, 59, 65, 66, 71, 73, 76, 80	17	Branches: 19, 20, 31, 35, 36, 37, 41, 46, 54, 58, 59, 65, 66, 71, 73, 76, 80
System load	-	1250.8 MW, 336.4 MVAr	-	1250.8 MW, 336.4 MVAr
Load bus	50	Allowable voltage range: [0.94–1.06]	50	Allowable voltage range: [0.94–1.06]

. In the basic IEEE 57-bus system, seven thermal generators are connected to buses 1, 2, 3, 6, 8, 9 and 12 and bus 1 is the balancing node [32].

In order to study the effect of PV power plants on the system, we built a modified IEEE 57-bus system with three PV plants, as shown in Figure 4. In the modified IEEE 57-bus system, four thermal generators connected to buses 1, 3, 8 and 12 are retained, and the other three thermal generators connected to buses 2, 6 and 9 are replaced by three PV power plants. The parameters of these three PV power plants are presented in

Table 2. The parameters of the three solar photovoltaic plants.

Plant	$R_{I.std}$ (W/m2)	Rated Power (MW)	The Parameters in Probability Density Function	Cost Factors ($/MW)
PV1	1000	100	$\phi =5.2$$; \gamma =0.6$	${\varpi }_d=1.7$$; {\varpi }_r=3.1$$; {\varpi }_p=1.5$
PV2			$\phi =5.1$$; \gamma =0.58$
PV3			$\phi =5$$; \gamma =0.62$

and their available power is shown in Figure 5.

5.2. Simulation Settings

Since the true optimal PF of the MOOPF is unavailable, the indicator Hypervolume (HV) is generally used to compare different algorithms. HV is described as below [33]:

$$HV\left(RP,PF\right)={\displaystyle \underset{x\in PF}{\overset{PF}{\cup }}\vartheta \left(RP,x\right)}$$

(35)

where $RP$ is the reference point $\left(1,1\right)$; $\vartheta$ is the hypercube size enclosed by $RP$ and $x$. This indicator can reflect an algorithm’s convergence and diversity. The higher the HV, the better the performance.

In order to test the workability of CMODE in tackling the MOOPF problem considering the uncertainty of PV power, it is compared with three peer methods, i.e., NSGA-II [34], CA-MOEA [14] and GrEA [35]. Their parameters settings are listed in

Table 3. Parameters settings of CMODE and other rivalries.

Method	Parameter Settings
CMODE	$N=200$$; F_m^G=0.1+0.9·rand$$; C{R}_m=rand$
NSGA-II	$N=200;$ mutation factor $P_M=1/D;$ crossover factor $P_C=1$ crossover parameters $D_M=20$$, D_C=20$
CA-MOEA	$N=200$$; P_C=1$$; P_M=1/D$$; D_M=20$$; D_C=20$
GrEA	$N=200$$; P_C=1$$; P_M=1/D$$; D_M=20$$; D_C=20;$ the number of divisions in each objective $DIV=45$

. To ensure the fairness, 30 independent trials were executed and the termination condition for each trial is $FE{\mathrm{s}}_{\max }=4\times {10}^4$.

5.3. Comparison of HV Results and Statistical Analysis

Table 4. Comparison of HV results.

Algorithm	Case 1		Case 2
	Mean	Std.	Mean	Std.
CMODE	0.07347	8.53 × 10−3	0.08413	7.67 × 10−3
NSGA-II	0.07241	5.99 × 10−3	0.07176	5.34 × 10−3
CA-MOEA	0.06316	6.03 × 10−3	0.07912	5.00 × 10−3
GrEA	0.07288	5.09 × 10−3	0.07719	9.45 × 10−3

shows, in both two cases the CMODE is the best one since its mean HV values are all higher than other methods. Although the standard deviation values of HV of CMODE are not the least.

Table 5. R⁻, p-value for the HV values of CMODE and other rivalries.

CMODE vs.	Case	R+	R−	p-Value	Sign
NSGA-II	Case 1	248	217	7.50 × 10−1	≈
	Case 2	450	15	0.80 × 10−5	+
CA-MOEA	Case 1	432	33	4.10 × 10−5	+
	Case 2	373	92	3.85 × 10−3	+
GrEA	Case 1	241	224	8.61 × 10−1	≈
	Case 2	361	104	8.20 × 10−3	+

provides the outcomes of Wilcoxon’s rank-sum test between CMODE and its rivalries at the 0.05 confidence level to reveal the statistical difference. The null hypothesis is that there is no significant difference between CMODE and the compared algorithms. The significance level is 0.05, and the hypothesis is rejected if the p-value is less than or equal to 0.05, i.e., there is a significant difference between the two algorithms compared. If the p-value is greater than 0.05, the hypothesis is accepted, i.e., there is no significant difference between CMODE and the compared algorithms. The signs “+” and “≈”represent that CMODE is signally superior or similar to its rivalries. Although MODE is statistically similar to NSGA-II and GrEA in case 1, the $R^{+}$ value is higher than the $R^{-}$ value. In the case 2, CMODE is the best. Overall, the comparison indicates that CMODE outperforms its rivalries.

5.4. Comparison of PF

We define the PF with the highest HV as the best PF, and conversely the PF with the lowest HV as the worst PF. The best and worst PF curves of all methods in these two cases are graphed in Figure 6 and Figure 7, respectively. In Figure 6a, the points of GrEA are more dispersed than CMODE, indicating it has a better diversity, but its convergence is inferior to CMODE. In Figure 6b, GrEA and CA-MOEA have a rich diversity and the convergence of NSGA-II is poor, the diversity and convergence of CMODE are in the mid. In Figure 7a, the diversity of CMODE is the best. In Figure 7b, CMODE has a richer diversity and the convergence of GrEA is the worst.

The difference between the best PF and the worst PF offered by CMODE is illustrated in Figure 8. In Figure 8a, the convergence in two scenarios is almost the same, but the best PF has a richer diversity signally. In Figure 8b, the diversity of the best PF is better than the worst signally.

5.5. Comparison of Solutions

Table 6. The comparison of part of solutions in case 1.

Method	Situation	Cost ($/h)	Emission (t/h)	$P_{loss}$ (MW)	$V_D$ (p.u.)
CMODE	Best Comp	42,378.25740	1.29978	14.39902	1.08593
	Best Cost	41,829.54006	1.60733	15.44529	1.17172
	Best Emission	45,553.66027	1.08585	16.51269	1.13773
NSGA-II	Best Comp	42,514.14538	1.28064	14.29878	1.06271
	Best Cost	41,835.31986	1.70183	16.62134	1.05436
	Best Emission	45,562.55097	1.08711	17.78190	1.05475
CA-MOEA	Best Comp	42,300.31554	1.49821	14.64055	1.12854
	Best Cost	41,829.85245	1.73326	16.56421	1.15373
	Best Emission	46,741.88170	1.18792	19.42758	1.10973
GrEA	Best Comp	42,095.55055	1.39710	15.70217	1.19019
	Best Cost	41,810.32857	1.74843	16.72726	0.93812
	Best Emission	46,334.09802	1.09455	18.51497	0.92364

and

Table 7. The comparison of part of solutions in case 2.

Method	Situation	Cost ($/h)	Emission (t/h)	$P_{loss}$ (MW)	$V_D$ (p.u.)
CMODE	Best Comp	31,510.96410	1.15458	14.92952	0.97206
	Best Cost	30,778.96877	1.51445	16.06183	0.99820
	Best Emission	34,763.02046	0.97188	18.75483	0.93152
NSGA-II	Best Comp	31,878.11249	1.10791	14.73706	1.20800
	Best Cost	30,960.60974	1.32659	16.33062	1.28074
	Best Emission	34,584.57320	0.97317	17.36800	1.23202
CA-MOEA	Best Comp	31,363.89771	1.18054	14.38301	1.17212
	Best Cost	30,860.00681	1.43874	16.46837	1.11257
	Best Emission	34,804.96768	0.97553	17.17602	1.18253
GrEA	Best Comp	31,528.49233	1.16682	16.37542	1.15121
	Best Cost	30,858.11783	1.58188	19.11846	1.10638
	Best Emission	34,888.15443	0.97865	17.70621	1.02394

give the results of three scenarios including the best compromise solution, the best cost solution, and the best emission offered by all methods in these two cases, respectively. Table 6 shows that in the three scenarios, GrEA offers the minimum cost 41,810.32857 $/h and CMODE yields the minimum emission 1.08585 t/h. With regard to the best compromise solution, according to the operation indicators, NSGA-II acquires the best one and CMODE is slightly better than CA-MOEA and GrEA. Table 7 indicates that the minimum cost is 30,778.96877 $/h and the minimum emission is 0.97188 t/h, where both data are provided by CMODE. For the best compromise solution of COMDE, the power loss is 14.92952 MW and the voltage deviation is 0.97206 p.u., which are slightly better than GrEA. Overall, CMODE can provide highly competitive solutions in these two cases. Moreover, the detail solutions offered by CMODE are provided in

Table A1. Best solutions offered by CMODE in case 1.

Case 1		Min	Max	Best Comp	Best Cost	Best Emission
Variable	PTG1 (MW)	0	575.88	186.3772	148.6171	223.4129
	PTG2 (MW)	30	100	100.0000	93.6162	100.0000
	PTG3 (MW)	42	140	70.8060	43.9203	140.0000
	PTG4 (MW)	30	100	99.8049	99.8664	100.0000
	PTG5 (MW)	165	550	364.2674	425.0176	287.8140
	PTG6 (MW)	30	100	100.0000	96.7985	99.9572
	PTG7 (MW)	123	410	343.9435	358.4092	316.1286
	V1 (p.u.)	0.95	1.1	1.0493	1.0533	1.0470
	V2 (p.u.)	0.95	1.1	1.0443	1.0456	1.0449
	V3 (p.u.)	0.95	1.1	1.0441	1.0453	1.0457
	V6 (p.u.)	0.95	1.1	1.0563	1.0462	1.0429
	V8 (p.u.)	0.95	1.1	1.0575	1.0424	1.0480
	V9 (p.u.)	0.95	1.1	1.0323	1.0277	1.0295
	V12 (p.u.)	0.95	1.1	1.0306	1.0437	1.0400
	T19 (p.u.)	0.90	1.10	0.9759	0.9864	1.0058
	T20 (p.u.)	0.90	1.10	1.1000	1.0354	1.0652
	T31 (p.u.)	0.90	1.10	0.9580	0.9952	1.0279
	T35 (p.u.)	0.90	1.10	1.0059	1.0155	0.9929
	T36 (p.u.)	0.90	1.10	0.9655	0.9359	0.9724
	T37 (p.u.)	0.90	1.10	1.0472	1.0311	0.9682
	T41 (p.u.)	0.90	1.10	0.9860	0.9923	0.9732
	T46 (p.u.)	0.90	1.10	0.9485	0.9738	0.9507
	T54 (p.u.)	0.90	1.10	0.9260	0.9170	0.9249
	T58 (p.u.)	0.90	1.10	0.9922	0.9830	1.0198
	T59 (p.u.)	0.90	1.10	0.9731	0.9567	0.9621
	T65 (p.u.)	0.90	1.10	0.9746	0.9616	1.0190
	T66 (p.u.)	0.90	1.10	0.9430	0.9636	0.9000
	T71 (p.u.)	0.90	1.10	1.0269	0.9983	1.0310
	T73 (p.u.)	0.90	1.10	1.0412	1.0067	0.9675
	T76 (p.u.)	0.90	1.10	0.9000	1.0132	1.0535
	T80 (p.u.)	0.90	1.10	1.0484	1.0409	1.0417
	QTG1 (MVAr)	−140	200	52.1131	61.7561	27.0031
	QTG2 (MVAr)	−17	50	28.0585	20.3215	37.9586
	QTG3 (MVAr)	−10	60	26.9675	41.2500	24.8212
	QTG4 (MVAr)	−8	25	12.7597	4.6859	−2.4807
	QTG5 (MVAr)	−140	200	52.3377	12.5234	50.0058
	QTG6 (MVAr)	−3	9	5.2316	8.1439	6.4411
	QTG7 (MVAr)	−150	155	54.2682	83.7349	97.1273
Objective	Emission (t/h)			1.2998	1.6073	1.0859
	Cost ($/h)			42,378.2574	41,829.5401	45,553.6603
Operation indicator	$V_D$ (p.u.)			1.0859	1.1717	1.1377
	$P_{loss}$ (MW)			14.3990	15.4453	16.5127

and

Table A2. Best solutions offered by CMODE in case 2.

Case 2		Min	Max	Best Comp	Best Cost	Best Emission
Variable	PTG1 (MW)	0	575.88	185.1707	132.3190	235.7068
	PPV1 (MW)	0	100	100.0000	100.0000	99.7895
	PTG2 (MW)	42	140	80.9551	51.1500	140.0000
	PPV2 (MW)	0	100	100.0000	100.0000	100.0000
	PTG3 (MW)	165	550	360.4079	433.4232	289.8099
	PPV3 (MW)	0	100	100.0000	100.0000	100.0000
	PTG4 (MW)	123	410	339.1958	349.9696	304.2488
	V1 (p.u.)	0.95	1.1	1.0166	1.0226	1.0109
	V2 (p.u.)	0.95	1.1	1.0128	1.0190	1.0081
	V3 (p.u.)	0.95	1.1	1.0240	1.0155	1.0214
	V6 (p.u.)	0.95	1.1	1.0390	1.0365	1.0374
	V8 (p.u.)	0.95	1.1	1.0472	1.0507	1.0555
	V9 (p.u.)	0.95	1.1	1.0195	1.0220	1.0278
	V12 (p.u.)	0.95	1.1	1.0236	1.0277	1.0406
	T19 (p.u.)	0.90	1.1	1.0382	1.0077	0.9963
	T20 (p.u.)	0.90	1.1	0.9816	0.9998	1.0451
	T31 (p.u.)	0.90	1.1	1.0515	1.0510	0.9962
	T35 (p.u.)	0.90	1.1	1.0412	0.9847	1.0599
	T36 (p.u.)	0.90	1.1	0.9137	0.9595	0.9000
	T37 (p.u.)	0.90	1.1	0.9975	0.9760	0.9929
	T41 (p.u.)	0.90	1.1	0.9784	0.9857	1.0127
	T46 (p.u.)	0.90	1.1	0.9733	0.9496	0.9693
	T54 (p.u.)	0.90	1.1	0.9114	0.9109	0.9005
	T58 (p.u.)	0.90	1.1	0.9608	0.9420	0.9773
	T59 (p.u.)	0.90	1.1	0.9623	0.9402	0.9293
	T65 (p.u.)	0.90	1.1	0.9550	0.9766	0.9586
	T66 (p.u.)	0.90	1.1	0.9648	0.9585	0.9395
	T71 (p.u.)	0.90	1.1	0.9644	0.9970	0.9782
	T73 (p.u.)	0.90	1.1	1.0470	0.9901	1.0281
	T76 (p.u.)	0.90	1.1	0.9827	0.9926	0.9920
	T80 (p.u.)	0.90	1.1	0.9893	0.9876	0.9844
	QTG1 (MVAr)	−140	200	16.9478	36.1716	−12.7746
	QPV1 (MVAr)	−17	50	20.7363	38.8423	22.0496
	QTG2 (MVAr)	−10	60	38.8840	15.6417	16.3274
	QPV2 (MVAr)	−8	25	3.9737	−3.1597	−6.3003
	QTG3 (MVAr)	−140	200	60.3842	56.9724	79.0276
	QPV3 (MVAr)	−3	9	4.0140	3.5384	5.4230
	QTG4 (MVAr)	−150	155	90.3729	91.3263	147.6832
Objective	Emission(t/h)			1.1546	1.5145	0.9719
	Cost ($/h)			31,510.9641	30,778.9688	34,763.0205
	$C_T$ ($/h)			30,251.7905	29,519.7952	33,504.8575
	$C_V$ ($/h)			1259.1736	1259.1736	1258.1630
Operation indicator	$V_D$ (p.u.)			0.9721	0.9982	0.9315
	$P_{loss}$ (MW)			14.9295	16.0618	18.7548

in the appendix.

In order to further verify the reliability of the solutions, the bus voltages of all the three scenarios in these two cases are graphed in Figure 9 and Figure 10, respectively. Figure 9a and Figure 10a present the generator bus voltage plots, i.e., voltage control variables plots. Figure 9b and Figure 10b present the load bus voltage plots, i.e., voltage state variables plots. In Figure 9 and Figure 10, all the bus voltages are within the limits, further indicating the effectiveness of CMODE in achieving feasible solutions in different scenarios. In both two cases, the voltages of buses 18, 25, 29 and 51 are closer to the upper boundary, indicating that overvoltage may occur in these buses. This is mainly because these buses are closer to the power sources. On the other hand, the voltages of 20, 34 and 57 are closer to the lower boundary, indicating that insufficient reactive power may occur in these buses and the main reason is that these buses are far away from the power sources.

In addition, the active power outputs (as shown in Figure 11a) of thermal generators and PV power plants and the corresponding emissions (as shown in Figure 11b) and operating costs (as shown in Figure 11c) provided by CMODE in the three scenarios for cases 1 and 2. Since the operating cost of PV power plants is relatively low and they do not emit polluting gases, so both the operating cost and emission in case 1 are consistently higher than those in case 2. To be specific, the introduction of PV power plants can reduce the total operating cost by 26.42% for the best cost solution scenario. On the other hand, the introduction of PV plants can also reduce the total emission by 10.50% for the best emission solution scenario. This is also the main reason for the massive use of solar energy. Table 6 and Table 7 also indicate that the PV power can reduce the operating cost and emission significantly.

6. Conclusions and Future Work

A MOOPF model considering the solar uncertainty is constructed, and a method CMODE is developed to better address the MOOPF model. The effectiveness of CMODE is validated by comparing the simulation outcomes with other methods in two cases. According to the results of HV, it is clear that the PFs offered by CMODE are overall better. From the results of the best solutions of three scenarios in these two cases, CMODE also provides competitive feasible solutions. With the penetration of PV power plants in power system, the operating cost and the emission are reduced by 26.42% and 10.50%, respectively. Therefore, the solar PV power brings significant economic and environmental benefits.

In this work, only the PV power uncertainty is considered. In future work, we plan to consider the PV self-consumed electricity and use prediction methods to obtain a more accurate and effective PV power model. Furthermore, we intend to consider more uncertainties including wind and loads in the MOOPF model. Moreover, some techniques such as reinforcement learning and hybridization will be integrated to further boost CMODE.